Medieval Islamic patterns: Penrose, eat your heart out

Medieval tiling patterns used a quasiperiodic technique that wasn't …

Patterning and periodic structures are very important in physics. This led to a tremendous amount of work on learning what shapes can tile an area or fill a volume. It was found that these shapes must fulfill certain symmetry properties. Only shapes that have two (rectangle), three (equilateral triangle), four (square), and six-fold (hexagon) rotational symmetries can tile a surface. This shape then becomes the unit cell of the pattern, which is created by translating copies of the shape by fixed integer multiples of the size. One notable finding, made in the early days of crystallography, was that shapes with five and ten-fold symmetries could not tile a surface or fill a volume.

However, examination of medieval Islamic decorations, called girih patterns, reveal plenty of five and ten-fold symmetric patterns that, apparently, do tile a surface. For the simpler patterns, the trick is apparent: these patterns are put together so that a combination of them yields a symmetry group that can tile by itself. This new, enlarged, unit cell is then used to tile the surface. These designs were thought to be made by a using compass and straight edge, which for the simpler patterns is certainly possible. However, many of the patterns are very complicated and cannot be broken into groups that fit into a symmetry group with simple tiling properties. It is possible that the whole surface could have been patterned entirely using a compass and straight edge, however, that would be cumbersome and error prone. The surviving patterns show no evidence of the geometric distortions that would be characteristic of using a compass and straightedge.

A possible solution has been discovered in a series of shapes found by Roger Penrose in 1974. These shapes fit together to tile a plane, yet they can have five and ten-fold symmetry. These shapes are not translated by an integer multiple of their size; instead, they are translated by the golden ratio of 1.618. Because these tiles never repeat exactly, they are called quasiperiodic patterns and have caught the attention of researchers studying girih patterns. The researchers discovered that if two shapes, called a kite and a dart, were scored with two lines each and placed in an almost repeating pattern, then the complicated girih patterns of medieval Islam were automatically formed. Although these patterns and their properties required complicated mathematics to describe, these simple tiles do not. The tiles are very simple and can be constructed using a compass and straight edge. After the tiles have been made, arbitrarily complicated patterns can be generated very rapidly by replicating the tiles. Not only that, the patterns constructed from these tiles are nearly perfect—certainly more perfect than could be made from a compass and straightedge.

A nice combination of modern mathematics describing some ancient beauty.

Chris Lee / Chris writes for Ars Technica's science section. A physicist by day and science writer by night, he specializes in quantum physics and optics. He lives and works in Eindhoven, the Netherlands.